Sculpted object turns into a cube when I enter edit mode
You more than likely have a Multiresoultion modifier on your object. You can switch to the Modifiers tab and press Apply on the modifier to apply this and see how your object looks in edit mode but if you apply it now, you will lose the ability to adjust any settings on the modifier and you might be left with a really high poly object which is undesirable if you decide to tweak the sculpt later (reduce view levels etc to reduce the load on the viewport).

If you still decide to apply the modifier, I would recommend making a copy of the sculpt just in case.

To keep the modifier and export the sculpt, when using the Collada exporter, look in the properties to the left and tick Apply modifiers.